where \(c_\alpha =\sup _{\zeta \ne z}|\psi _1(\zeta )-\psi _1(z)|/|\zeta -z|^\alpha \) . Taking the limit \(r\rightarrow (1-|z|)\)  we obtain
\(|\partial \psi (z)|\leqslant c_\alpha (1-|z|)^{\alpha -1}\) , see also [1]} or [2]}. This means that
\(\partial \psi \in L_p(\mathbb {D})\)  if \((1-\alpha )p<1\) . Similarly, the \(L_p\) –integrability of the derivative \(\overline{\partial }\psi =\overline{\psi _2^{\prime }}\)  is established under the same condition on \(p\) . The lemma is proved.
